The Interaction of Alpha Particles and a Neutron Pair Within a Nucleus

Suppose that within a nucleus the neutrons and protons form alpha particles whenever possible.
The formation of an alpha particle entails a binding energy of 28.295 million electron volts (MeV).
The binding energy from the formation of alpha particles accounts for most of the binding energies of
nuclides. If BE is the binding energy of a nuclide and a is the number of alpha particles it contains then
the excess binding energy XSBE is given by

XSBE = BE − 28.295*a

Below is shown the plot of the XSBE for the nuclides which could contain an integral number of alpha particles,
hereafter referred to as the alpha nuclides.

A bent line in which the bends come at the points where the number of neutrons (and the number of protons) is equal to the magic numbers of {2, 6, 14, 28}. The statistical fit is near perfect, as
shown by the coefficient of determination (R²) for the regression of 0.999169.

The plot of the excess binding energies of the nuclides which could contain an integral number of
alpha particles plus an additional neutron pair is shown below.

The bends in this case are not so sharp as in the case of the alpha nuclides. This is because the
effects of transitions to a new neutron shell and a new proton shell is spread over two changes in the
number of alpha particles.

At this point it is important to note that the crucial variable is the number of neutrons rather than
the number of alpha particles. A comparison of the excess binding energies for the alpha nuclides
and the alpha+2neutron cases is shown below.

A larger picture of the differences is useful at this point.

Note that the jumps in the level occur at the magic numbers.

The differences can be estimated by a broken-line regression function of the following form